Rate of Prefix-free Codes in LQG Control Systems

TL;DR

This paper establishes bounds on the data rate needed for prefix-free coding in LQG control systems, linking information theory with control performance and providing computable bounds via SDP.

Contribution

It derives a lower bound on the rate for prefix-free codes in LQG control and provides an upper bound using a practical coding scheme, with a quantifiable gap between them.

Findings

Lower bound matches the infimum of a directed information expression.

Upper bound constructed with quantizer and Shannon-Fano coding.

Gap between bounds is less than 0.754r+1 bits per step.

Abstract

In this paper, we consider a discrete time linear quadratic Gaussian (LQG) control problem in which state information of the plant is encoded in a variable-length binary codeword at every time step, and a control input is determined based on the codewords generated in the past. We derive a lower bound of the rate achievable by the class of prefix-free codes attaining the required LQG control performance. This lower bound coincides with the infimum of a certain directed information expression, and is computable by semidefinite programming (SDP). Based on a technique by Silva et al., we also provide an upper bound of the best achievable rate by constructing a controller equipped with a uniform quantizer with subtractive dither and Shannon-Fano coding. The gap between the obtained lower and upper bounds is less than $0.754r+1$ bits per time step regardless of the required LQG control performance, where $r$ is the rank of a signal-to-noise ratio matrix obtained by SDP, which is no greater than the dimension of the state.